Pure (sub)module (in $M$)

Question

What is the difference between a pure module $P$ and a pure module $P$ in a module $M$ ? I have a slight intuition though, what is the relationship between purity and solvability of equations.

Answer 1

As Bernard says, there is no such thing as a pure module because it is a property of submodules.

In response to the second part of your question, consider the two different definitions of purity, the first using model theoretic notions which we will call pp-pure.

Definition Let $R$ be a ring and $M$ is a left $R$-module. If $L$ is a submodule of $M$, then the embedding $L\to M$ is pure if for every pp-formula $\phi$ of length $n$, one has $$\phi(L)=L^{n}\cap\phi(M)$$ Here $\phi(M)=\{m\in M^{n}:M\models\phi(m)\}$.

This condition states that the inclusion preserves and reflects solutions to pp-formulae.

Now consider the following algebraic definition, which we will call $\otimes$-pure:

Definition If $R$ is a ring and $M$ is a left $R$-module, then a submodule $L\subseteq M$ is pure if for every right $R$-module $T$ the sequence $$0\to T\otimes_{R}L\to T\otimes_{R}M$$ is exact.

So here is the relationship between the two:

Proposition pp-purity and $\otimes$-purity are the same

I won't give a proof here, but one can be found in $\S 2.4$ of these notes by Mike Prest.

To talk about solutions of equations, we need pure injective.

Definition An $R$-module $M$ is pure injective if and only if every system of equations with parameters in $M$ that is finitely solvable in $M$ is solvable in $M$.

There is also an algebraic definition of pure-injectivity, which is being injective over pure submodules.

Now, given any $R$-module $M$, there is a pure injective module $PI(M)$ and pure embedding $M\to PI(M)$ that is the 'smallest' pure injective module containing $M$ purely, called the pure injective envelope of $M$. Because $M\to PI(M)$ is a pure embedding, it reflects solutions to pp-formulae, all of which have solutions in $PI(M)$ (provided they're finitely solvable) as it is pure injective.

Obviously, if $M$ is pure-injective to begin with it is its own pure-injective hull and all systems of equations equations that are finitely solvable have solutions in it. As an example, any injective module is pure-injective, as is any finitely generated module over a complete local ring. Obviously there are many more.

An extensive description of purity and its role in the model theory of modules can be found in Mike Prest's book Purity, Spectra and Localisation.